2024年美国国际大学生数学建模竞赛(MCM+ICM)题目+翻译.doc

IMPORTANT CHANGE TO CONTEST RULES FOR MCM/ICM 2024: Teams (Student or Advisor) are now required to submit an electronic copy (summary sheet and solution) of their solution paper by email to .Your email MUST be received at COMAP by the submission deadline of 8:00 PM EST, February 13, 2024. Teams are free to choose between MCM Problem A, MCM Problem B or ICM Problem C.COMAP Mirror Site: For more in: MCM: The Mathematical Contest in ModelingICM: The Interdisciplinary Contest in Modeling 2024 Contest Problems MCM PROBLEMS PROBLEM A: The Leaves of a Tree How much do the leaves on a tree weigh? How might one estimate the actual weight of the leaves (or for that matter any other parts of the tree)? How might one classify leaves? Build a mathematical model to describe and classify leaves. Consider and answer the following: Why do leaves have the various shapes that they have? Do the shapes “minimize” overlapping individual shadows that are cast, so as to maximize exposure? Does the distribution of leaves within the “volume” of the tree and its branches effect the shape? Speaking of profiles, is leaf shape (general characteristics) related to tree pro structure? How would you estimate the leaf mass of a tree? Is there a correlation between the leaf mass and the size characteristics of the tree (height, mass, volume defined by the profile)?In addition to your one page summary sheet prepare a one page letter to an editor of a scientific journal outlining your key findings. 2024美赛A题：一棵树的叶子（数学中国翻译） “一棵树的叶子有多重？”怎么能估计树的叶子（或者树的任何其它部分）的实际重量？怎样对叶子进行分类？建立一个数学模型来对叶子进行描述和分类。模型要考虑和回答下面的问题： 为什么叶子具有各种形状？ 叶子之间是要将相互重叠的部分最小化，以便可以最大限度的接触到阳光吗？树叶的分布以及树干和枝杈的体积影响叶子的形状吗？ 就轮廓来讲，叶形（一般特征）是和树的轮廓以及分枝结构有关吗？ 你将如何估计一棵树的叶子质量？叶子的质量和树的尺寸特征（包括和外形轮廓有关的高度、质量、体积）有联系吗？除了你的一页摘要以外，给科学杂志的编辑写一封信，阐述你的主要发现。PROBLEM B: Camping along the Big Long River Visitors to the Big Long River (225 miles) can enjoy scenic views and exciting white water rapids. The river is inaccessible to hikers, so the only way to enjoy it is to take a river trip that requires several days of camping. River trips all start at First Launch and exit the river at Final Exit, 225 miles downstream. Passengers take either oar- powered rubber rafts, which travel on average 4 mph or motorized boats, which travel on average 8 mph. The trips range from 6 to 18 nights of camping on the river, start to finish. The government agency responsible for managing this river wants every trip to enjoy a wilderness experience, with minimal contact with other groups of boats on the river. Currently, X trips travel down the Big Long River each year during a six month period (the rest of the year it is too cold for river trips). There are Y camp sites on the Big Long River, distributed fairly uniformly throughout the river corridor. Given the rise in popularity of river rafting, the park managers have been asked to allow more trips to travel down the river. They want to determine how they might schedule an optimal mix of trips, of varying duration (measured in nights on the river) and propulsion (motor or oar) that will utilize the campsites in the best way possible. In other words, how many more boat trips could be added to the Big Long Rivers rafting season? The river managers have hired you to advise them on ways in which to develop the best schedule and on ways in which to determine the carrying capacity of the river, remembering that no two sets of campers can occupy the same site at the same time. In addition to your one page summary sheet, prepare a one page memo to the managers of the river describing your key findings. 问题B：沿长江大露营到Big Long River（225英里）游玩的游客可以享受那里的风景和振奋人心的急流。远足者没法到达这条河，唯一去的办法是漂流过去。这需要几天的露营。河流旅行始于First Launch，在 Final Exit结束，共225英里的顺流。旅客可以选择依靠船桨来前进的橡皮筏，它的速度是4英里每小时，或者选择8英里每小时的摩托船。旅行从开始到结束包括大约6到18个晚上的河中的露营。负责管理这条河的政府部门希望让每次旅行都能尽情享受野外经历，同时能尽量少的与河中其他的船只相遇。当前，每年经过Big Long河的游客有X组，这些漂流都在一个以6个月长短的时期内进行，一年中的其他月份非常冷，不会有漂流。在Big Long上有Y处露营地点，平均分布于河廊。随着漂流人数的增加，管理者被要求应该允许让更多的船只漂流。他们要决定如何来安排最优的方案：包括旅行时间（以在河上的夜晚数计算）、选择哪种船（摩托还是桨船），从而能够最好地利用河中的露营地。换句话说，Big Long River在漂流季节还能增加多少漂流旅行数？管理者希望你能给他们最好的建议，告诉他们如何决定河流的容纳量，记住任两组旅行队都不能同时占据河中的露营地。此外，在你的摘要表一页，准备一页给管理者的备忘录，用来描述你的关键发现。ICM PROBLEMPROBLEM C: Modeling for Crime Busting Click the title below to download a ZIP the 2024 ICM Problem.Your ICM submission should consist of a 1 page Summary Sheet and your solution cannot exceed 20 pages for a maximum of 21 pages.2024 ICM ProblemModeling for Crime BustingYour organization, the Intergalactic Crime Modelers (ICM), is investigating a conspiracy to commit acriminal act. The investigators are highly confident they know several members of the conspiracy, buthope to identify the other members and the leaders before they make arrests. The conspirators andthe possible suspected conspirators all work for the same company in a large office complex. Thecompany has been growing fast and making a name for itself in developing and marketing computersoftware for banks and credit card companies. ICM has recently found a small set of messages from agroup of 82 workers in the company that they believe will help them find the most likely candidates forthe unidentified coconspirators and unknown leaders. Since the message traffic is for all the officeworkers in the company, it is very likely that some (maybe many) of the identified communicators in themessage traffic are not involved in the conspiracy. In fact, they are certain that they know some peoplewho are not in the conspiracy. The goal of the modeling effort will be to identify people in the officecomplex who are the most likely conspirators. A priority list would be ideal so ICM could investigate,place under surveillance, and/or interrogate the most likely candidates. A discriminate line separatingconspirators from nonconspirators would also be helpful to distinctly categorize the people in eachgroup. It would also be helpful to the DAs office if the model nominated the conspiracy leaders.Before the data of the current case are given to your crime modeling team, your supervisor gives youthe following scenario (called Investigation EZ) that she worked on a few years ago in another city. Eventhough she is very proud of her work on the EZ case, she says it is just a very small, simple example, butit may help you understand your task. Her data follow:The ten people she was considering as conspirators were: Anne#, Bob, Carol, Dave*, Ellen, Fred,George*, Harry, Inez, and Jaye#. (* indicates prior known conspirators, # indicate prior known nonconspirators)Chronology of the 28 messages that she had for her case with a code number for the topic of eachmessage that she assigned based on her analysis of the message:Anne to Bob: Why were you late today? (1)Bob to Carol: That darn Anne always watches me. I wasnt late. (1)Carol to Dave: Anne and Bob are fighting again over Bobs tardiness. (1)Dave to Ellen: I need to see you this morning. When can you come by? Bring the budget files. (2)Dave to Fred: I can come by and see you anytime today. Let me know when it is a good time. Should Ibring the budget files? (2)Dave to George: I will see you later lots to talk about. I hope the others are ready. It is important toget this right. (3)Harry to George: You seem stressed. What is going on? Our budget will be fine. (2) (4)Inez to George: I am real tired today. How are you doing? ( 5)Jaye to Inez: Not much going on today. Wanna go to lunch today? (5)Inez to Jaye: Good thing it is quiet. I am exhausted. Cant do lunch today sorry! (5)George to Dave: Time to talk now! (3)Jaye to Anne: Can you go to lunch today? (5)Dave to George: I cant. On my way to see Fred. (3)George to Dave: Get here after that. (3)Anne to Carol: Who is supposed to watch Bob? He is goofing off all the time. (1)Carol to Anne: Leave him alone. He is working well with George and Dave. (1)George to Dave: This is important. Darn Fred. How about Ellen? (3)Ellen to George: Have you talked with Dave? (3)George to Ellen: Not yet. Did you? (3)Bob to Anne: I wasnt late. And just so you know I am working through lunch. (1)Bob to Dave: Tell them I wasnt late. You know me. (1)Ellen to Carol: Get with Anne and figure out the budget meeting schedule for next week and help mecalm George. (2)Harry to Dave: Did you notice that George is stressed out again today? (4)Dave to George: Darn Harry thinks you are stressed. Dont get him worried or he will be nosing around.(4)George to Harry: Just working late and having problems at home. I will be fine. (4)Ellen to Harry: Would it be OK, if I miss the meeting today? Fred will be there and he knows thebudget better than I do. (2)Harry to Fred: I think next years budget is stressing out a few people. Maybe we should take time toreassure people today. (2) (4)Fred to Harry: I think our budget is pretty healthy. I dont see anything to stress over. (2)END of MESSAGE TRAFFICYour supervisor points outs that she assigned and coded only 5 different topics of messages: 1) Bobstardiness, 2) the budget, 3) important unknown issue but assumed to be part of conspiracy, 4) Georgesstress, and 5) lunch and other social issues. As seen in the message coding, some messages had twotopics assigned because of the content of the messages.The way your supervisor analyzed her situation was with a network that showed the communicationlinks and the types of messages. The following figure is a model of the message network that resultedwith the code for the types of messages annotated on the network graph.Figure 1: Network of messages from EZ CaseYour supervisor points out that in addition to known conspirators George and Dave, Ellen and Carolwere indicted for the conspiracy based on your supervisors analysis and later Bob selfadmitted hisinvolvement in a plea bargain for a reduced sentence, but the charges against Carol were later dropped.Your supervisor is still pretty sure Inez was involved, but the case against her was never established.Your supervisors advice to your team is identify the guilty parties so that people like Inez dont get off,people like Carol are not falsely accused, and ICM gets the credit so people like Bob do not have theopportunity to get reduced sentences.The current case:Your supervisor has put together a networklike database for the current case, which has the samestructure, but is a bit larger in scope. The investigators have some indications that a conspiracy is takingplace to embezzle funds from the company and use internet fraud to steal funds from credit cards ofpeople who do business with the company. The small example she showed you for case EZ had only 10people (nodes), 27 links (messages), 5 topics, 1 suspicious/conspiracy topic, 2 known conspirators, and 2known nonconspirators. So far, the new case has 83 nodes, 400 links (some involving more than 1topic), over 21,000 words of message traffic, 15 topics (3 have been deemed to be suspicious), 7 knownconspirators, and 8 known nonconspirators. These data are given in the attached spreadsheet files:Names.xls, Topics.xls, and Messages.xls. Names.xls contains the key of node number to the officeworkers names. Topics.xls contains the code for the 15 topic numbers to a short description of thetopics. Because of security and privacy issues, your team will not have direct transcripts of all themessage traffic. Messages.xls provides the links of the nodes that transmitted messages and the topiccode numbers that the messages contained. Several messages contained up to three topics. To helpvisualize the message traffic, a network model of the people and message links is provided in Figure 2.In this case, the topics of the messages are not shown in the figure as they were in Figure 1. These topicnumbers are given in the and described in Topics.xls.Figure 2: Visual of the network model of the 83 people (nodes) and 400 messages between thesepeople (links).Requirements:Requirement 1: So far, it is known that Jean, Alex, Elsie, Paul, Ulf, Yao, and Harvey are conspirators.Also, it is known that Darlene, Tran, Jia, Ellin, Gard, Chris, Paige, and Este are not conspirators. Thethree known suspicious message topics are 7, 11, and 13. There is more detail about the topics in fileTopics.xls. Build a model and algorithm to prioritize the 83 nodes by likelihood of being part of theconspiracy and explain your model and metrics. Jerome, Delores, and Gretchen are the seniormanagers of the company. It would be very helpful to know if any of them are involved in theconspiracy.Requirement 2: How would the priority list change if new information comes to light that Topic 1 isalso connected to the conspiracy and that Chris is one of the conspirators?Requirement 3: A powerful technique to obtain and understand text information similar to thismessage traffic is called semantic network analysis; as a methodology in artificial intelligence andcomputational linguistics, it provides a structure and process for reasoning about knowledge orlanguage. Another computational linguistics capability in natural language processing is text analysis.For our crime busting scenario, explain how semantic and text analyses of the content and context ofthe message traffic (if you could obtain the original messages) could empower your team to developeven better models and categorizations of the office personnel. Did you use any of these capabilities onthe topic descriptions in to enhance your model?Requirement 4: Your complete report will eventually go to the DA so it must be detailed and clearlystate your assumptions and methodology, but cannot exceed 20 pages of write up. You may includeyour programs as appendices in separate files that do not count in your page restriction, but includingthese programs is not necessary. Your supervisor wants ICM to be the worlds best in solving whitecollar,hightech conspiracy crimes and hopes your methodology will contribute to solving importantcases around the world, especially those with very large databases of message traffic (thousands ofpeople with tens of thousands of messages and possibly millions of words). She specifically asked youto include a discussion on how more thorough network, semantic, and text analyses of the messagecontents could help with your model and recommendations. As part of your report to her, explain thenetwork modeling techniques you have used and why and how they can be used to identify, prioritize,and categorize similar nodes in a network database of any type, not just crime conspiracies and messagedata. For instance, could your method find the infected or diseased cells in a biological network whereyou had various kinds of image or chemical data for the node